In general, a thin structure is characterized as having weak resistance to bending force and strong resistance to stretching force, such that movement of a thin structure is solved by a stiff equation (a type of differential equation). When compressive force is applied to a thin structure, contraction occurs. As the compressive force is increased to pass a critical point, the thin structure is vertically bent, also known as buckling. Buckling is a deformation which occurs abruptly so that it is a very unstable reaction. Therefore, the simulation of that kind of buckling results in a divergence problem in differential equations for simulating the movement of cloth.
The buckling instability refers to the contrast of the decrease in compressive force and increase in bending deformation. This structural instability makes the system matrix extremely ill-conditioned or indefinite, and in a case where the time period of the simulation steps is increased, the system matrix becomes divergent. In conventional simulations of the movement of cloth, a model in which the instability of buckling exists is used such that it is very difficult to simulate a phenomenon of wrinkles (buckling) forming on the surface of the cloth.
The buckling instability problem arises not from the stiff equation itself but from the structural instability of the cloth. Therefore, simply employing an implicit method cannot solve the problem. Resolution of the buckling instability problem was conventionally sought by adding a damping term to a system matrix. However, although the addition of the damping term can stabilize the system, it obstructs the naturalness of the movement of the cloth. In other words, the damping force prevents formation of wrinkles on the cloth's surface, and prevents wrinkles from disappearing. Hence, a simple addition of a damping term is not desirable to simulate the movement of cloth.
Furthermore, a continuous body model used in the past brought forth an undesirable result due to the following reasons. A coarse discretization should be allowed to guarantee a reasonable performance in computer graphic applications. However, a continuous body model requires a very fine mesh in order to simulate large deformation of cloth. Therefore, a reasonable processing speed cannot be obtained. Another drawback is that the continuous body model cannot properly deal with the divergence derived from buckling, thereby requiring additional calculations. [Eischen et al. 1996] uses a non-linear shell model for cloth simulation and performs a finite element analysis to simulate buckling. However, great care and cautious measures have to be undertaken such as arc length control to prevent divergence due to the non-linearity of the load-deflection curve or the singularity of the instantaneous stiffness matrix caused by buckling.
In [Baraff and Witkin 1998], a system of connected triangles was proposed as a model. The in-plane deformation energy of each triangle is derived from the continuous mechanism. The bending deformation is based on the angle between adjacent triangles. Therefore, the bending deformation and the in-plane deformation are modelled independently, which allows a large bending deformation between the-triangles regardless of the in-plane rigidity of each triangle, thereby resolving the drawback of the continuous body model. However, the buckling instability still remains in this model because each triangle is modeled as an almost incompressible material and the bending rigidity between triangles is very weak.
Still furthermore, in order to simulate the hysteresis characteristics of the movement of cloth, the results of measuring the hysteresis characteristics of the cloth have been directly used. However, a hysteretic response curve obtained by the measurement can be applied only when a curve change during the measurement and a curve change during the simulation are the same, such that it is not generally appropriate for dynamic simulation. In other words, a hysteretic response curve is a function of the entire curvature history, such that it is not reasonable to simulate by measuring the curve instead of by constructing a physical model for the hysteresis characteristics.